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ESCC 2016, July 10-16, Athens, Greece

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Accelerating Benders Decomposition

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ESCC 2016, July 10-16, Athens, Greece

  1. 1. Fakhri Ashkan, Ghatee Mehdi, Fragkogios Antonios, Saharidis K.D. Georgios UNIVERSITY OF THESSALY School of Engineering Department of Mechanical Engineering Division of Production Management & Industrial Administration 3rd International Conference on Energy, Sustainability and Climate Change, July 10-16, 2016 With the contribution of the LIFE programme of the European Union - LIFE14 ENV/GR/000611
  2. 2.  BENDERS METHOD  PROPOSED ALGORITHM  APPLICATION-RESULTS  CONCLUSION
  3. 3.  Jacobus Franciscus (Jacques) Βenders: Dutch Mathematician (1925-today) Emeritus Professor of Operations Research at the Eindhoven University of Technology.  Benders, Jacques F. “Partitioning procedures for solving mixed-variables programming problems.” Numerische mathematik 4.1 (1962): 238–252.
  4. 4.  Since 1962, more than 5.000 papers have been published, which modify, extend and accelerate the method.  Sections:  Mixed-Integer Linear Programming,  Stochastic Programming,  Multi-Objective Programming,  Non-Linear Programming  Etc.
  5. 5.  Applications:  Crew Scheduling,  Plant Scheduling,  Supply Chain Network Design,  Power Systems  Etc.
  6. 6.  Mixed-Integer Problem  Relaxed Master Problem (RMP)  Primal Subproblem (PSP) Complicating variables
  7. 7. START Solve Master Problem. Compute Lower Bound. Solve SubProblem. Upper Bound – Lower Bound >= e Produce Benders Optimality Cut. END YES NO SubProblem is Feasible. Compute Upper Bound. NO YES Produce Benders Feasibility Cut. Add Benders Cuts to Master Problem.
  8. 8.  BENDERS METHOD  PROPOSED ALGORITHM  APPLICATION-RESULTS  CONCLUSION
  9. 9.  In order to apply Benders decomposition method, the duality of the subproblem is necessary (Dual Subproblem exists).  The Primal Subproblem has to be continuous and no duality gap exists.  Application at Mixed-Integer Linear Problems:  Master Problem: Integer variables and  Primal Subproblem: Only continuous variables of the original problem.
  10. 10.  However, the decomposition of the original problem might lead to a Primal Subproblem, which includes integer variables.  The application of the classical Benders decomposition method is impossible.  A Branch-and-Cut algorithm is proposed, which allows the use of Benders method at the case of integer subproblem.
  11. 11.  The integrality constraints of the Primal Subproblem are relaxed and the Relaxed Subproblem is solved in a Branch-and- Bound framework. Integer Primal Subproblem (PSP) Relaxed Primal Subproblem (RPSP)
  12. 12. 0 1 2 5 4 6 3 87 109 i i+1 i+2 Branch-and-Bound
  13. 13. 0 1 2 5 4 6 3 87 109 i i+1 i+2 Branch-and-Cut (with Benders method) Application of Benders method between RMP and A-RPSP
  14. 14. Augmented Relaxed Primal Subproblem (A-RPSP(k)) Relaxed Master Problem (RMP)
  15. 15. 0 1 2 5 4 6 3 87 109 i i+1 i+2 Branch-and-Cut (with Benders method) Application of Benders method between MP and RPSP MP0 SP0 MP0 SP1 MP0 SP2 MP0 SP3 MP0 SP4 MP0 SP5 MP0 SP6 MP0 SP7 MP0 SP8 MP0 SP9 MP0 SP10
  16. 16. Proof that the Benders cuts, which are produced in a node of the Branch-and-Bound tree are valid for all descendant nodes, but not necessarily for the non- descendant nodes. These cuts, referred to as LOCAL CUTS, can be used to warm start the master problem of each descendant node leading to better initial bounds. GENERAL CUTS are formed out of the LOCAL CUTS. This general form enables us to reuse the generated (local) cuts in the whole tree by updating some values of the function.
  17. 17. Augmented-RMP (A-RMP(k))
  18. 18. 0 1 2 5 4 6 3 87 109 Branch-and-Cut (with Benders method+Local Cuts) MP0 SP0 MP1 SP1 MP1 SP2 MP3 SP3 MP3 SP4 MP5 SP5 MP5 SP6 MP7 SP7 MP7 SP8 MP9 SP9 MP9 SP10 MP1=MP0+BendersCuts0 MP3=MP1+BendersCuts2 MP5=MP3+BendersCuts4 MP7=MP5+BendersCuts5 MP9=MP7+BendersCuts7 “Local Cuts”
  19. 19. Generalization of Local Cuts (General Cuts) if we are going to use the generalized cuts (55) or (57) in another node, say k2, it is sufficient to replace y* by the optimal value of y at node k2
  20. 20. Global Cuts if we are going to use the generalized cuts (55) or (57) in another node, say k2, it is sufficient to replace y* by the optimal value of y at node k2
  21. 21.  BENDERS METHOD  PROPOSED ALGORITHM  APPLICATION-RESULTS  CONCLUSION
  22. 22.  Capacitated Fixed Charge Multiple Knapsack Problem- CFCMKP.  Pure Integer Problem  Binary variables  Integer Primal Subproblem APPLICATION No.1
  23. 23.  20 Examples  Solution with the classical Branch-and-Cut algorithm with Benders method, but without use of “Local Cuts” (B&C-NotLC) and with the proposed algorithm with “Local Cuts” (B&C-LC).  Comparison of the results.
  24. 24. Instances B&C-NotLC B&C-LC Relative difference B&C-LC over B&C-NotLC Α/Α M N CPU time (Sec.) Iterations CPU time Iterations CPU time Iterations (Sec.) 1 5 40 5.54 380 2 59 -63.90% -84.50% 2 5 60 11.15 593 3.27 71 -70.70% -88.00% 3 5 80 3.2 186 1.94 26 -39.40% -86.00% 4 6 80 8.81 375 4.4 77 -50.10% -79.50% 5 7 80 3.72 129 1.75 29 -52.90% -77.50% 6 8 80 31.79 838 0.48 11 -98.50% -98.70% 7 9 80 4.91 185 1.55 24 -68.50% -87.00% 8 10 50 334.29 3431 179.11 638 -46.40% -81.40% 9 10 60 62.64 1642 2.41 53 -96.10% -96.80% 10 10 70 264.7 2582 11.72 175 -95.60% -93.20% 11 10 80 411.44 3477 2.51 34 -99.40% -99.00% 12 10 90 1.23 49 0.76 9 -38.40% -81.60% 13 10 100 25.53 474 4.82 48 -81.10% -89.90% 14 15 50 13.25 391 9.39 92 -29.10% -76.50% 15 15 60 6.23 189 3.59 64 -42.40% -66.10% 16 15 70 3.29 95 2.33 39 -29.20% -58.90% 17 15 80 54.63 892 14.52 91 -73.40% -89.80% 18 15 90 13.01 301 8.39 76 -35.50% -74.80% 19 15 100 1565.1 4404 0.35 7 -99.90% -99.80% 20 15 110 36.78 502 6.5 57 -82.30% -88.60%
  25. 25.  Significant reduction of the CPU time (-29.1% ως -99.9%, Average: -64.7%)  Large reduction of the total number of iterations of Benders algorithm inside all nodes (-58.9% ως-99.8%, Average: -84.9%). The small number of iterations is due to the better initial lower bounds. Thus, the MP is solved fewer times and the proposed algorithm is faster.
  26. 26.  Environmental Multi-Modal Journey- Planning Problem- E-MMJP (Action B.3)  Pure Integer Problem (Binary variables-Integer Primal Subproblem)  Mixed Integer Linear Program (MILP) in order to compute the optimal journey between the departure and arrival stops of the public network.  In-between those two stations, the model prompts the user to use up to a number of different modes of transport, depending on his/her input. While in the network, the user follows an optimal journey that minimizes the travel time APPLICATION No.2
  27. 27.  Decision variables  Xi,j,k,n Binary Variable used to represent whether a transfer is made from i to j with mode k and trip n.  Si,j,k,n Non-negative general integer variable used to represent the departure time from i to j with k and n.
  28. 28.  Objective Function Minimization of 2 criteria, the total environmental cost and the total travel time of the journey, which is proposed to the user. Coefficients a and b are predefined by the user. The Environmental Cost Ci,j,k is pre-computed for each arc i-j and mode k of the public transportation network using emission calculation models, that take into consideration several parameters, such as the type of fuel (gasoline, diesel, electricity etc.) and the fuel consumption, which concern the vehicle of the public means of transport. Other parameters concern the trip, such as the distance and the gradient between the stops , , , , ,, , 1 1 1 1 ( * * )* N N M L i j k i j k ni j k i j k n Min z a bC TT X= = = = = +∑∑∑∑
  29. 29.  Due to the pure integrality of the problem, when decomposed using Benders Decomposition method, the Primal Subproblem will be an integer one.  Thus, this modelling approach for the environmental MMJP problem should be based on the proposed Branch-and-Cut algorithm using Benders Decomposition with Local Cuts (B&C-LC)
  30. 30.  BENDERS METHOD  PROPOSED ALGORITHM  APPLICATION-RESULTS  CONCLUSION
  31. 31.  A novel, better and valid algorithm is proposed for the solution of problems, where after application of Benders decomposition, the Primal Subproblem is integer.  Acceleration of the classical Branch-and-Cut algorithm with the application of “Local Cuts”.
  32. 32. Thank you for your attention With the contribution of the LIFE programme of the European Union - LIFE14 ENV/GR/000611

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